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DNCERNING  COMPACT  KURSCHAK  FIELDS 


A  DISSERTATI(3N 

QUITTED   TO   THE   FACULTY   OF  THE   OGDEN   GRADUATE   SCHOOL  OF   SCIENCE   IN 
CANDIDACY  FOR   THE   DEGREE   OF  DOCTOR   OF   PHILOSOPHY 
DEPARTMENT   OF  MATHEMATICS 


BY 


VISHNU  DATTATREYA  GOKHALE 


Private  Edition,  Distributed  By 

The  University  of  Chicago  Libraries 

Chicago,  Illinois 


Reprinted  from 

The  American  Journal  of  Mathematics 

Vol.  XLIV,  No.  4,  October,  1922 


^be  inmveretti?  of  (tbtcaoo 


CONCERNING  COMPACT  KURSCHAK  FIELDS 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY  OF  THE  OGDEN  GRADUATE  SCHOOL  OF  SCIENCE  IN 

CANDIDACY  FOR  THE  DEGREE  OF  DOCTOR  OF   PHILOSOPHY 

DEPARTMENT  OF  MATHEMATICS 


VISHNU  DATTATREYA  GOKHALE 


Private  Edition,  Distributed  By 

The  University  of  Chicago  Libraries 

Chicago,  Illinois 


Reprinted  from 

American  Journal  of  Mathematics 

Vol.  XLIV,  No.  4,  October,  1922. 


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CONCERNING   COMPACT  KURSCHAK  FIELDS. 

By  Vishnu  Dattatreya  Gokhale. 

CONTENTS. 

I.    Introduction 298 

1.  Definition  of  a  field,  characteristic 298,  299 

2.  Algebraic  closure,  algebraic  extensions 299 

3.  Definition  of  a  KiirscMk  field 300 

4.  Perfection,  Kiirschdk's  results 300 

II.    Compactness  and  Algebraic  Extensions 302 

1,  2.    Compactness,  compactness  and  perfection 302 

3.  Compactness  and  algebraic  closure 303 

4.  5.    Hensel-Klirschdk  fields,  Ostrowski's  results,  applications 304,  306 

6.        Compactness  and  adjunction  of  algebraic  elements 306 

III.  The  Hensel-Kurschak  Field  3cs 308 

1,  2.  The  Hensel-Kiirschdk  field  X3 308,  310 

3,  4.  Perfection  and  compactness  of  Xg 310,  311 

5.  Subfield  $R(g,  x)  of  X^ 312 

6.  Adjunction  of  elements  algebraic  with  respect  to  g 313 

IV.  The  Properties  p,  A,  P,  cpt 314 

1-5.     Complete  existential  theory  of  these  four  properties 314 

Abstract. — A  Kiirschdk  field  is  afield  with  a  modulus  ("bewertete  Korper),  in  the  sense 
defined  by  Kiirschak  in  his  memoir,  "Ueber  Limesbildung  und  die  allgemeine  Korper- 
theorie"  (Crelle,  vol.  142,  1913).  This  modulus  plays,  in  the  general  field,  eesentially  the 
same  role  as  the  absolute  value  in  the  fields  of  classical  analysis,  viz.,  real  number  system, 
complex  number  system,  etc.  Kiirschak  proves  that  every  Kiirschak  field  determines 
(in  the  sense  of  isomorphism)  a  definite  algebraically  closed  and  perfect  Kiirschdk  super- 
field,  the  smallest  such  superfield,  being  (in  the  sense  of  isomorphism)  a  subfield  of  every 
such  superfield.  This  definite  superfield  is  the  {smallest)  algebraically  closed  and  perfect 
extension  of  the  original  Kiirschak  field. 

In  the  present  paper  the  author  sets  up  the  notion  compactness.  This  notion  is 
analogous  to  M.  Frechet's  compactness  and  to  the  J-compactness  in  E.  H.  Moore's  General' 
Analysis.  It  is  a  generalization  of  the  following  property  in  the  point  set  theory:  Every 
infinite  set  of  points  in  a  bounded  domain  has  at  least  one  condensation  point.  He  then 
studies  the  properties  of  algebraically  closed  and  compact  fields,  and  compactness  under 
the  adjunction  of  algebraic  elements.  Using  Ostrowski's  results  he  proves  the  theorem 
that  the  smallest  algebraically  closed  extension  of  a  compact  field  is  compact  if,  and  only  if, 
it  can  be  obtained  by  adjoining  a  single  algebraic  element.  The  last  part  of  the  paper 
develops  a  complete  existential  theory  of  the  four  properties:  (1)  of  characteristic  other 
than  zero,  (2)  algebraic  closure,  (3)  perfection,  and  (4)  compactness.  Out  of  the  2*  =  16 
possibilities  11  are  shown  to  be  existent  and  the  remaining  5  non-existent. 

297 


529:^95 


298  GoKHALE :  *  Vdhderning  Compact  Kiirschdk  Fields. 

I.  Introduction. 

1.1.  Field.— Following  Moore,*  H.  Weber,t  and  Steinitz4  a  field  is 
defined  in  the  following  manner:  § 

D  "[  1  Jt  ^    C<jU   .  _1_   on'.5P'Pto:iS.1.2.6.6.     \y    on!5p>pto5P.3.4.5.7\ 

where  ^  =  [[p]  is  a  general  class  of  elements  p{pi,  P2,  etc.);  +,  X  are 
single-valued  functions,  +  (pi,  P2)  is  a  definite  element  p  denoted  by 
Pi  +  Pi,  etc. ;  and  the  properties  numbered  1-7  are : 


(Pi,  P2,  Pz) 


(1)  Pi  +  (P2  +  Ps)  =  {pi  +  P2)  +  Ps ' 

(2)  Pl  +  P2  =   P2+  Pi 

(3)  IP1P2)P3  =   PliP2Pz) 

(4)  P1P2  =  P2P1 

(5)  pi(p2  +  Pz)  =  P1P2  +  PiPz 

Here  (pi,  p2,  Ps)  indicates  that  the  equations  (1-5)  hold  for  every  choice  of 
the  elements  pi  p2  pz  of  the  class  '^. 

(6)  Pi-P2') '3l  p^pi  + p=  P2 

Here   the  meaning  of  the  logical  signs  * ) ',  3,1,  and   '   will  be   clear 
from  the  following  reading  of  the  property: — For  every  pi  and  p2,  there 
exists  uniquely  an  element  p  such  that  pi  +  p  •=  P2. 
Hence, 

(60)  3 1  2^" :  9  :  p  :  )  '.  p -\-  z  =  p-pz  =  z. 

Here  the  notation  2^'  reads  "z  belongs  to  the  class  *>|3."  This  unique  element 
z  is  called  the  zero  eleTnent  of  the  field. 

(7)  3  p  7^  2  :  pi  7^  Z-P2  • )  •  3l  p  ?  pip  =  P2, 

to  be  read  "  There  exsists  an  element  p  different  from  the  element  z  (of  60), 
and  for  every  such  element  pi  and  every  element  p2  there  exists  uniquely  an 
element  p  such  that  pip  =  P2." 

*E.  H.  Moore,  "A  Doubly  Infinite  System  of  Simple  Groups"  (Chicago  Congress, 
1893,  pp.  208-242),  p.  210.     This  paper  will  be  referred  to  as  M. 

t  H.  Weber,  "Die  allgemeinen  Grundlagen  der  Galoischen  Gleichungstheorie "  (Mathe- 
matische  Annalen,  43,  1893,  pp.  521-549),  p.  526;  also  "Algebra"  (II  edition,  1898),  Vol.  I, 
p.  492. 

t  Steinitz,  "  Algebraische  Theorie  der  Korper"  (Crelle,  137,  1910,  pp.  167-309),  p.  172. 
This  paper  wUl  be  referred  to  as  S. 

§  For  postulational  definitions  of  a  field  see : 

L.  E.  Dickson,  "Definitions  of  a  Group  and  a  Field  by  Independent  Postulates" 
{Trans.  A.  M.  S.,  Vol.  6,  1905,  pp.  198-204),  p.  202. 

E.  V.  Huntington,  "Note  on  the  Definitions  of  Abstract  Groups  and  Fields  by  Sets  of 
Independent  Postulates"  {Trans.  A.  M.  S.,  Vol.  6,  pp.  181-197),  pp.  186,  191.  This  paper 
also  contains  a  bibliography. 


Gokhale:   Concerning  Compact  Kiirschdk  Fields.  299 

This  property  can  be  easily  seen  to  be  equivalent  to  the  following: 

(7')        3  p^  z  :  3u^'  ^\j>')  '  pu=  p  :p  ^  z')  -3  p'  epp'  =  u]. 

Also,  we  have 

(7o)  3l  u^'  ^  p  '  )  '  pu  ='p':  pi  9^  Z'pipx  =  z  :  )  :  Pi  =  z. 

This  element  u  is  called  the  unit  element  of  the  field,  and  the  element  (which 
can  be  easily  seen  to  be  unique)  p'  associated  with  p  in  the  second  part  of 
7'  is  called  the  reciprocal  of  p. 

Hereafter  we  denote  the  elements  p  of  the  class  ''P  of  a  field  not  by  p 
but  by  /  (/i, /2,  etc.). 

1.11.  Characteristic. — A  field  of  characteristic  p,  in  notation  %^  is  defi ned 
in  the  following  manner:  * 

D  1.2  %^  \  =  \  %  '  ^  '  3  n  ^  nu  =  z- p  =  the  smallest  such  n. 

By  using  the  properties  of  the  field  it  can  be  proved  that  in  this  case, 

(1)  p  is  a  prime. 

(2)  /  5^  z  :  )  :  n/  =  z  •  ~  '  n  =  a  multiple  of  p. 

On  the  other  hand,  for  every  positive  integer  m  and  a  prime  p,  there  existsf 
one  and  only  one  finite  field,  the  so-called  Galois  field  [_p^~\,  with  char- 
acteristic p. 

Fields  without  any  such  characteristic  are  said  to  be  of  characteristic 
zero,  in  notation  %^.  Henceforth  in  the  notation  f^^  we  shall  understand 
p  to  be  an  indefinite  prime  number.     Thus  every  field  is  of  the  type  '^^  or  %'p. 

1.2.  Algebraic  closure,  algebraic  extensions. — An  algebraically  closed 
field  5,  in  notation  f^"*,  is  thus  defined:  | 

Z)  1  3      ^^''  ^  •  '^  :  3  : 

n-{hh,  •  •  -,fn)  :  )  :  3  f  ^  Up  +  /i/""^  +...+/,  =  ^. 

An  element  j  algebraic  with  respect  to  i^,  in  notation  j^^  *  is  defined  as:  § 

2)1.4    j^^'^':^  ■:{],%)  :5:3  =  -^n-(/i,  -  -  - ,  fn)  '  ^ ' 

This  unique  integer  n  is  called  the  order  of  j.     Here  x  is  an  indeterminate, 
and  j  belongs  to  a  field  %'  containing  %  as  a  subfield,  in  notation,  %'  d  %. 
Steinitz  has  shown]  |  how  to  extend  a  field  by  the  adjunction  of  an 

*  S,  p.  181.  Cf.  also:  J.  Konig: — Einleitung  in  die  allgemeine  Theorie  der  algebraischen 
Grossen  (Leipzig,  1913),  p.  408.  Konig  uses  the  terms  "orthoid"  and"  pseudoorthoid"  for 
fields  of  characteristic  zero  and  p  respectively. 

tM,  p.  211. 

X  S,  p.  260. 

§  S,  p.  183. 

II  S.  p.  197. 


300  GoKHALE :    Concerning  Compact  Kiirschdk  Fields. 

algebraic  element  j.  This  extension  of  a  field  ^,  in  notation  ^(j),  is  the 
extension  in  the  sense  that  it  is  the  smallest  and  unique  (in  the  sense  of 
isomorphism).  He  also  shows  how  to  get  the  extension  which  reduces 
completely  a  given  polynomial  <p  irreducible  in  ^.      Steinitz  also  proves:* 

Thm  1.1  e  :  )  :  3  W^'  :  ^  :  g'^-r^"^ ' )  '  %"  o  %\ 

This  field  %',  unique  in  the  sense  of  isomorphism,  we  denote  by  \^a: 
the  (smallest)  extension  of  %  having  the  property  A. 

Every  field  ^  has  ojie  and  only  one  prime  suhfield  f  '^o',  Vo  is  isomorphic 
with  the  rational  number  system  or  the  integral  number  system  taken 
modulo  p,  according  as  ^5  is  of  characteristic  0  or  p.  By  absolute  algebraic 
field,  in  notation  '^,  we  mean  the  algebraically  closed  extension  of  such 
a  prime  field  '^q  J 

A  field  i^',  algebraic  extension  of  ^,  in  notation  %  ^^^^  is  thus  defined  as:  § 

D  1.5  W^'^' :  ^  :  (g',  5)  :  ^  :  S'  3  g./'"'  '  )  '/"'''- 

Note:  The  negative  sign  denotes  the  absence  of  the  property.  Thus 
the  last  implication  should  be  read:  Iff  does  not  belong  to  %,  it  is  algebraic 
with  respect  to  %. 

1.3.  Kiirschak  Field. — A  Kiirschak  field,  in  notation  9^^,  is  thus  defined:  || 

Z>1.6  9i^(l5;||       ||on5to.:-*'-°.1.2.3.4)^ 

where  i5  denotes  a  field,  51"^®*^-°  the  class  of  all  real  non-negative  numbers, 
and  the  properties  1-4  of  the  single-valued  function  1 1  1 1  (which  will  here- 
after be  called  the  modulus),  are: — 

(1)  /=z-~-  11/11  =  0 

(2)  IIMll  =  |l/illll/.||     .-.      \\«\\  =  l\,f    f. 

(4)  3/:,:||/||^0-|l/||^l     • 

Hereafter  we  designate  the  elements  of  9t  not  by  /  but  by  k{ki,  k^,  etc.). 

1.4.  Limit,  Perfect  Extension. — A  limit  k  oi  a  sequence  ki,  k^,  •••  in 
notation  {kn},  is  thus  defined:  Tj 

D  1.7  Lkn=k\=\{k,    [kn])  ^L\\k-kn\\   =  0. 


*  S,  p.  287. 
t  S,  p.  180. 

I  S,  p.  199. 
§  S,  p.  198. 

II  J.  Ktirschdk,  "Uber  Limesbildung  und    allgemeine  Korpertheorie"    {Crelle,    142, 
1913,  pp.  211-233),  p.  211.     This  paper  will  be  referred  to  as  K. 

UK,  p.  222. 


GoKHALE :    Concerning  Compact  Kiirschak  Fields.  301 

As  an  immediate  consequence  of  the  definition, 

Thm  1.2        Lkn=  k-LK=  k'  :)  :  k  =  k' -  \\k\\  =  \\k'\\  =  L\\kn\\. 

n  n  n 

The  usual  properties  of  limits,  for  instance:  if  a  sequence  has  a  limit, 
every  subsequence  has  the  same  limit  and  conversely;  the  sum  of  limits 
equals  the  limit  of  the  sum;  etc.,  follow  in  the  usual  manner. 

A  Cauchy  sequence  {kn},  or  {kn}  satisfying  the  Cauchy  condition:  in 
notation  {kn}"-"  :    is: 

D  1.8     {kn}'-'\=  i  {kn]  :^\e  :)  :3ne-^-n>ne.)  .\\kn-  knA\  <e. 

Here,  as  further,  e  is  a  real  positive  number.     As  an  immediate  consequence, 

we  have 

Thm  1.3  3Lkn  ' )  "  {A;^)^-''. 

n 

The  converse  however  is  not  true;  e.g.,  in  the  field  of  rational  numbers 
with  the  absolute  value  as  modulus,  not  all  Cauchy  sequences  have  rational 
limits.  A  field  dlo  or  a  subclass  9?o  (not  necessarily  a  field)  of  a  field  9?  for 
which  the  converse  holds  is  called  perfect:  in  notation  dio.  Thus  the  defini- 
tion of  perfection  is:  * 

2)1.9  9??:=i9?o''^:3  :  {konV'" '  )  '  3  ko  ^  L  kon  =  ko; 

n 

in  this  definition  (as  also  further,  where  necessary)  we  denote  the  elements 
of  9^0  by  ko{koi,  fco2,  etc.). 

By  a  method  analogous  to  G.  Cantor's  in  building  up  the  real  number 
system  from  the  rational  numbers,  Kiirschak  has  shown f  how  to  extend  a 
Kiirschak  field  so  as  to  make  it  perfect.  The  smallest  such  extension 
(unique  in  the  sense  of  isomorphism)  of  aJR  we  designate  notationally  by 
^^ :  the  (smallest)  extension  of  di  having  the  property  P. 

Given  a  9?^  and  j^^^,  Kiirschak  defines  ||  j  ||  so  that  9?(j)  is  a  Kiirschak 
field:  X  in  notation  9J(j)^.  Defining  in  dl^  the  modulus  of  every  element 
algebraic  with  respect  to  di  in  the  same  manner,  we  have  9?^.  Making 
this  perfect,  we  get  9?/p,  in  the  sense  (9?a)p.  Kiirschak  then  shows§  that 
di^p  is  algebraically  closed.     If  we  start  with  any  Kiirschak  field  9f,  we 

*  K,  p.  228. 

t  K.,  p.  228. 

JK,  p.  245.  Kiirschdk  defines  \\j\\  to  be  ||/„|1^'»,  where  /„  is  the  /„  in  D  1.4.  A. 
Ostrowski  in  his  "Tiber  sogennante  perfecte  Korper"  {Crelle,  147,  1917,  pp.  191-204), 
p.  196,  and  "Uber  einige  Losungen  der  Functionalgleichung  <p{xy)  =  <f>(x)<p(y)"  (Acta 
Mathematica,  41,  1918,  p.  271-284),  p.  280,  has  shown  that  this  is  the  only  possible 
definition  of  ||  j|l. 

These  papers  will  be  referred  to  as  O2,  O3  respectively. 

§  K,  p.  251. 


302  Gokhale:   Concerning  Compact  Kurschak  Fields. 

must  first  make  it  perfect  and  then  follow  this  process.     Thus: 

Kiirschak's  final  theorem  is  thus:  f 

Thm  1.4  ^  :  )  :  ^Rifp  '^  •  ^'^-^-^^  ' )  '  ^'  ^  9?p^p 

This  perfect  and  algebraically  closed  extension  of  9^  we  shall  denote  by 
^AP  or  9^p^. 

II.  Compactness  and  Algebkaic  Extensions. 

2.1.  Compactness. — In  classical  point  set  theory  we  have  the  theorem 
that  every  infinite  set  of  points  in  a  bounded  domain  has  at  least  one  con- 
densation point.  A  similar  property  for  a  general  class  for  which  the  limit 
function  exists  is  defined  by  Frechet.|  Classes  having  this  property  are 
said  to  be,  according  to  his  nomenclature,  compact.  The  following  defini- 
tion of  compactness  in  the  case  of  a  Kurschak  field  is  naturally  suggested 
as  analogous  to  this  definition  of  Frechet  and  the  definition  of  J-compactness 
in  Moore's  theory.  § 

A  subset  (not  necessarily  a  field)  of  a  Kurschak  field  is  said  to  he  compact 
if  every  infinite  sequence  of  elements  of  the  set  such  that  the  (smallest)  upper 
bound  of  the  modulus  is  finite,  contains  at  least  one  Cauchy  subsequence  with 
its  limit  element  in  the  set. 
In  notation: 

^  ^'^  •  )  •  3    {h,    {/l^}P^--°°-*'^-)   '  L  hn^  =    ^0. 

Here  as  in  7)  1.9  9^o  is  a  subset  not  necessarily  a  field  of  a  Kiirschak  field 
di;  and  {rim}  is  a  properly  monotonic  increasing  sequ£nce  of  positive  integers 
n.  This  property  denoted  notationally  in  the  above  definition  as  pr.  mon,  inc. 
is  thus  defined: 

D  2.11  {ri,„}P^-'"»°'°''- :  =  i  {nm}  :  '  :  mi  >  mz  •  )  '  n^,  >  nm,- 

2.2.  Compactness  and  Perfection. — We  now  prove  that: 
Thm  2.1  .  W^'-)'diE- 

*  In  his  theses,  "Uber  einige  Fragen  der  allgemeinen  Korpertheorie"  {Crelle,  143, 
1913,  pp.  255-284).  p.  284,  A.  Ostrowski  finds  under  what  conditions  this  step  is  necessary. 
He  also  shows  (p.  260)  that  the  algebraically  closed  extension  of  Hensel's  p-adic  numbers 
is  not  perfect.     This  paper  will  be  referred  to  as  Oi. 

tK,  p.  251. 

t  M.  Frechet,  "Sur  quelques  points  du  calcul  fonctionnel"  {Rendiconti  del  Circ.  Math, 
d.  Palermo,  22,  1906),  p.  6. 

§E.  H.  Moore,  "Lectures  on  Matrices  in  General  Analysis"  (University  of  Chicago, 
1919-1920). 


Gokhale:    Concerning  Compact  Kilrschdk  Fields.  303 

Proof:  Consider  a  Cauchy  sequence  {kon}.  Since  it  is  a  Cauchy  se- 
quence, 

e  :)  :3nc^n>  ne')  '  \\  hn  —  hn,  ||  <  e. 

Therefore  for  such  ann,  \\  kon  \  \  Hes  between  |  1 1  kon^  1 1  —  ^  I  and  \  \  kon,  1 1  +  «• 

Hence  B  1 1  fcon  !  |  <  °o  •     Therefore  by  compactness  we  have  a  subsequence 

« 

having  a  Hmit  in  9?o-  Hence  the  original  sequence  has  the  same  limit  in 
dlo.     Hence  the  theorem. 

Note:  The  converse  of  this  theorem,  however,  is  not  true.  See  Thm 
2.3  below.     Compare  also  theorems  3.3  and  3.4  in  the  next  section. 

2.3.  Compactness  and  Algebraic  Closure. — 

Thm  2.2  di^'-'^'-a^O  :)  :  3k^  \\  k  \\  =  a. 

Proof:  The  theorem  is  obvious  when  a  =  0  or  1,  the  corresponding  k 
being  z  and  u  respectively.     When  a  is  neither  the  proof  is  as  follows : 

By  D  1.6  property  4,  there  exists  an  element  whose  modulus  is  neither  0 
nor  1 ,     Let  this  element  be  denoted  by  ko  and  let  its  modulus  be  ^. 

By  the   theory  of  the  real    number  system  3  {r„}  ^  jL  ^'■"  =  a,   where 

n 

{r„}  is  a  sequence  of  ordinary  rationals  r.     Hence  by  9^"*, 
3   {k'o''}-B\\k^''\\  <oo. 

Therefore  by  di'^\ 

3  (k,  {nm}'''-'^^''-^'-)  3  L  t-n.  =  k 

m 

and  I!  k  \\  =  L\\  kl"m  \\  =  L  ^'""m  =  a. 

m  m 

Using  Moore's  results,*  Steinitz  proves:  f 

Thm  2.3  Z^"  5^  2  :  )  :  3  m  ^/^^-i  -  u  =  z. 

We  shall  use  this  theorem  to  prove: 

Thm  2.4  9^A-«p* '  )  '  9^°. 

Proof:  We  shall  give  a  direct  proof  of  a  contrapositive  of  the  theorem, 
viz., 

^p-A  .  )   .  S)J-cpt^ 

Every  91^'^  contains  '^p,  the  absolute  algebraic  field  characteristic  p. 
Consider  { k^ }  ''''«*^'=*-  ^''.     By  theorem  2.3  we  have : 

n:):\\kn\\  =  0        or         ||  ^„  [h  =  i, 

where  q  =  p"^  —  1  in  theorem  2.3.     Hence  5  ||  ^n  11  =  1  .'.  <  oo.     But 


*  M,  p.  220. 
tS,  p.  251. 


304  GoKHALE :   Concerning  Compact  Kilrschak  Fields. 

this  sequence  cannot  have  a  Cauchy  subsequence;  for,  otherwise, 

e  :  )  :  3  (ni,  n^)^'"^''^  II  ^n^  -  ^nJI  <  2e, 

viz.,  ni,  n2  each  greater  than  ne  in  D  1.8.  But  from  theorem  2.3  and  the 
fact  that  ni9^  n2')'  \\  K,  \\  9^  \\  kn^  ||,  we  have 

ni9^  n2')'\\  kn,  -  kn,\\  =  1. 

Taking  e  <  1/2  we  get  a  contradiction.     Thus  this  particular  sequence  has 
no  Cauchy  subsequence.     Hence  the  theorem. 
We  have  further: 

Thm  2.5  dt"""")  -3  n' II  7m  II  5^  1. 

Proof:  Here  also  we  prove  the  contrapositive,  viz.. 

di^n-)-\\nu\\  =  1:)  :  ^-^p*. 

Since  II  nw  II  =  1  (n),  we  have  the  characteristic  zero,  and  ||  rw  ||  =  1  (r) 
where  r  is  any  ordinary  rational  different  from  zero.     Consider  { r„ }  ^*^"'''*. 

We  have  5  II  r„2t  II  =  1  .'.  <  00. 

n  , 

Now  the  sequence  {rnu]   cannot  have  any  Cauchy  subsequence;    for 
otherwise, 

e:)  :3  (ni,  na)*^"""*  ^  ||  rn,u  -  rn,u  \\  <  2e, 

but  since  niP^  n^')  '  r^u  —  rn^  =  {rn,  —  rn,)u  7^  z')'  \\  rn,u  —  Vn^u  ||  =  1, 
taking  e  <  1/2  we  get  a  contradiction.     Hence  the  theorem. 
We  have  also: 

Thm  2.6  dt'"'  • )  •  (the  first  n^\\nu\\9^  1)^'^^ 

Proof:    For   otherwise   if   w  =  nin2,   where  ?ii  ^  ng  <  n,   nu  =  nin^u 
=  {niu){n<2.u).     Hence  ||  tiw  |1  =  1  which  contradicts  the  hypotheses. 

2.4.  Hensel-Kurschak  Fields,  Ostrowski's  Results. — A  Kiirschak  field 
where  the  property  3  of  the  modulus  in  D  1.6  is  replaced  by  the  stronger 
property: 
(30  II  /i  +  /2  II  ^  greater  of  (||  /i  ||,  ||  f,  \\)     {f„  f,) 

is  called  a  Hensel-Kurschak  field:  in  notation  ©.     Thus: 

D  2.2  (S=(g;||        ||on!5to5l'eal-ao.l.2.3'.4), 

where  the  properties  1,  2,  4  are  those  in  Z)  1.6,  and  3'  is  defined  above. 
Hereafter  we  denote  the  elements  of  <S  by  h{li\,  hi,  etc.);  the  class  of  all 
Hensel-Kurschak  fields  will  be  denoted  by  H,  and  the  superscript  H  will 
denote  the  property  of  belonging  to  this  class.  These  notations  will  be 
used  only  when  it  is  necessary  to  emphasize  the  fact  that  the  Kiirschdk 
field  under  consideration  is  a  Hensel-Kurschak  field. 


Gokbale:   Concerning  Compact  Kurschak  Fields.  305 

The  modulus  of  a  Hensel-Kiirschak  field,  which,  when  necessary,  we  shall 
call  Hensel-Kiirschak  modulus,  is  what  Ostrowski*  calls  a  non-Archimedian 
modulus:  in  notation:   ||     ||non-arch_     jjg  g^jg^  proves: 

Thm\  2.7  II     ll^o'^-^'^i':  '^  :  n' )  '  \\nu\\  ^  I. 

Thmt2.8  \\h\\>  \\h,\\  :)  :|Ui  +  A2||  =  ||  Ai  ||. 

A  modulus  which  does  not  obey  this  stronger  property  3'  is  called 
Archimedian:  in  notation:  jj     ||^''^ 

A.  Ostrowski  has  investigated  real-valued  solutions  of  the  functional 
equations : 

(p{xy)  =  (p{x)(piy) 


(^    ^yatlonal 

<p{x-\-  y)  ^  ip{x)  -f  (p{y)\ 

His  conclusions,  modified  for  the  Kiirschak  modulus  by  the  fact  that  the 
modulus  is  non-negative,  are: 

rAm§2.9     ^0:):       (1)  r  5^  2  ' ) '•  ||  r  ||  =  1;  ||  k  ||  =  0 

or(2)r-)-||r||  =  jrl"        0<p^l 

or  (3)  r  • )  •  II  r  II  =  c"^-^^        0  <  c  <  1,  pP'^''"^ 

Here  0(2?,  r)  is  the  order  of  r  with  respect  to  p,  defined,  following  Hensel  ||  as: 


Z)  2.3  I  ^  =  -rP" '  (^'  ^yvimetop  I  . )  .  o(^^  r) 


V 


In  the  above  theorem,  the  modulus  is  Archimedian  in  case  2,  non-Archi- 
median in  the  other  two.  For  n  =  2  in  theorem  2.6,  there  does  exist  a 
field  with  Archimedian  modulus,  viz.,  glcompiex^  ^j^j^  ^^^  modulus  as  defined 
in  the  second  part  of  theorem  2.9.  If  we  denote  this  field  by  5Ip,  and  define 
equivalence:  in  notation  ~  :  as  isomorphism  between  Kurschak  fields 
which  is  preserved  under  the  limit  process,  we  have: 


Thm  2.10 

P1-P2 

■■^■■-^'--"  iitm- 

Ostrowski  prov 

es:1I 

Thm  2.11 

^-"■)'3p^di  c5r. 

and  finally,** 

Thm  2.12 

9fl-~a.P.A..y^  ^5JcompIex_ 

*  O3.  p.  273. 

t03,  p.  273. 

JOs.p.  274. 

§  O3,  p.  276. 

11  K.  Hensel,  ' 

'Theorie  der  algebraischen  Zahlen"  (Leipzig,  1908,  Vol.  I). 

IfOa,  p.  281. 

**  O3,  p.  282. 

306  GoKHALE :   Concerning  Compact  Kiirschak  Fields. 

2.5.  Deductions  from  Ostrowski's  Results. — rrom  theorem  2.7  the 
Archimedian  or  non-Archimedian  character  of  the  modulus  depends  upon 
the  moduli  of  the  elements  of  the  prime  subfield.     Hence  we  have: 

Thm  2.13  9^'  3  9?  : )  :  9^^^"^^ '  -  *  9e'^^-^>. 

Here  the  two  parentheses  go  together. 
From  theorem  2.12  we  have 

nm2.14  ^^-^•^■^••)-9l«p*.     " 

We  shall  now  prove  an  important  theorem  to  be  used  in  the  sequel: 
viz.,  compactness  is  not  extensionally  attainable*     To  be  more  precise, 

Thm  2.15  9fi-oi>t.p  • )  •  -  3  9fi'^^.cvt^ 

Proof:  Since  the  field  is  perfect  every  Cauchy  sequence  has  a  limit 
element  in  the  field.  Want  of  compactness,  therefore,  is  due  to  the  fact 
that  there  exists  an  infinite  sequence  which  has  no  Cauchy  subsequence. 
Now  every  extension  preserves  the  moduli;  this  fact  will  therefore  persist 
in  every  extension.     Hence  no  extension  will  make  the  field  compact. 

We  show  in  section  4  theorem  4.10o  that  ^~pf)',  where  ®  is  the  Hensel 
field  of  p-adic  numbers.  Since  in  this  case  compactness  is  not  extensionally 
attainable  (in  virtue  of  theorem  2.15),  we  have,  using  theorems  2.9  and  2.4, 

Thm  2.16  di^-  "^P*  :  ~  :  91  ~  glcompiex^ 

2.6.  Compactness  and  Adjimction  of  Algebraic  Elements. — An  algebraic 
element  of  the  first  kind:  in  notation  algi :  is  thus  defined :  f 

p  24    •^'''''  :^   ^    I   U,m-'  :f''  :  <p'-'^-™^"'-^'-  ■<pU)  =  z 

Note:  Here  the  dot  in  the  last  brackets  stands  for  the  argument  of  the 
function  \_x  —  j  |  a:]:  read  "a;  —  j  as  a:  varies." 

An  algebraic  element,  which  is  not  of  the  first  kind  is  said  to  be  of  the 
second  kind:  in  notation:  alg2.  Similar  definitions  hold  for  algebraic 
extensions  of  the  first  and  second  kind. 

Ostrowski  proves:  J 
nm2.17  9?^-i^^^^-)-9?0r. 

We  prove  the  analogous  theorem: 
Thm  2.18  g^cpt.jaig,  n.y  ^(^jyvt^ 

*  A  propertT/  is  said  to  be  extensionally  attainable  when  there  exists  an  extension  which 
has  that  property.  Compare  E.  H.  Moore,  "Introduction  to  a  Form  of  General  Analysis," 
Yale  University  Press,  1910,  pp.  53,  54. 

tS,  p.  231. 

t  Ox,  p.  275. 


Gokhale:   Concerning  Compact  Kiirschdk  Fields.  -      307 

Proof:  Let  m  be  the  order  of  j  and  fx  an  upper  bound  of  the  modulus  of 
an  infinite  sequence  {kn,  ij"*"^  -{-■••  +  kn,  m\  n}  oi  elements  of  9^(j).  We 
have  to  show  that  this  sequence  has  at  least  one  Cauchy  subsequence  with 
a  limit  element  in  di(j). 

Let  us  denote  this  sequence  by  {io,  n]  and  let  the  m  conjugates  of  io.n 
be  denoted  by  ii,  „,  22,  n,  •  •  •,  im,  n',  *i,  n  =  io,  n  (n).  Let  the  m  conjugates 
of  j  be  denoted  by  ji,  j-z,  "  • ,  jm',  j  ^  ji- 

In  the  normal  extension*  of  dt  which  contains  9?0)  we  have  for  every 
n  the  m  linear  equations: 

kn,  ijr'  +    •  •  •    +   ^-n.  ..   =   ^1.  n 
kn,  ij^r'  +    •  •  •   +  A;.,,  m  =   *2,  « 


/<^n.  ij",!.    ^  +    ■  •  ■   -^  kn.  m  =   im,  n. 

Since  j  is  of  the  first  kind,  the  determinant  of  coefficients  involving  j  and 
its  conjugates  is  not  the  zero  element.  Hence  we  can  solve  for  the  A:'s 
and  get  kn,  r  {r,  n)  as  linear  functions  of  the  i's  with  coeflScients  independent 
of  n.    Also  !|  ?r,  n  11  =  II  *o.  n  \\  (w,  r).     Hence 5  ||  A;„,  r  !|  <  00    (r).     Consider 

n 

now  the  sequence  [kn,  1};  by  Dt*'^*  and  the  result  proved  just  now,  this 
sequence  has  at  least  one  Cauchy  subsequence  with  a  limit  element  in  9?. 
Take  one  such  Cauchy  subsequence  and  the  corresponding  subsequence  of 
[id,  n\,  say  [i'n].  Denote  the  coefficients  of  the  different  powers  of  j  in 
this  sequence  by  prime  letters.  As  above  we  can  prove  that  [k'n,  2}  has  a 
Cauchy  subsequence.  Take  the  corresponding  subsequence  of  [i'n]  which 
is  itself  a  subsequence  of  the  original  sequence.  Continuing  this  process, 
after  a  finite  number  of  steps,  viz.,  m  steps,  we  get  a  subsequence  of  the 
original  sequence  such  that  the  coefficient  sequences  are  all  Cauchy  se- 
quences.    Let  this  final  sequence  be  denoted  by  [inh     Then  if 

in  =   kn,  li"*-^  + \-kn,m       (w), 

we  have: 

nr7l2   :  )    :  II  in,  -  in,  \\    ^  F(||  K,  1  -   K^  1  II,  -  '   II  K,  m  -   kn„  m  |i) 

■     F(llilh-S  •••,  liill,  1), 

where  gr  denotes  "the  greater  of."  Hence  this  subsequence  is  a  Cauchy 
sequence.  Using  theorems  2.1  and  2.17  we  see  that  the  limit  element  is  in 
$K(j).     Hence  the  theorem. 

.  An  algebraic  extension  is  said  to  be  finite  with  respect  to  the  original 
field:  in  notation  fin  :  when  it  can  be  obtained  by  adjoining  a  finite  number 
*  S.,  p.  207. 


308    •  Gokhale:   Concerning  Compact  Kurschdk  Fields. 

of  algebraic  elements  to  the  original  field :  * 

D  2.5     ^'«-^  !  ^  I  m',  $K)  :  ^  :  3n-{h,k,  ■ '  ^iO^'^'^  ^  ^'  =  ^^(ii,  •  •  ^jn). 

Steinitz  proves:  f 

Thm  2.19    di-  (ii,  •  •  • ,  jnr^^  ^'  : )  :  aj^'^'  ^  ^  9^0')  =  9?(ii.  '  • ' ,  in). 

Ostrowski  proves :  t 

TAm  2.20  di''  :)  :dif')-  diT'  ''■ 

Using  theorem  2.16,  he  then  proves§  a  theorem,  which,  in  view  of 
theorem  2.19,  we  can  state  as: 

Thm  2.21  $R^  : )  :  9?r  ~  •  ^j'""  ' '  9^0*)  =  ^a- 

We  prove  the  analogous  theorem: 
Thm  2.22  di'""  : )  :  dlT  '  ~  *  3/'^^  ^  di(j)  =  ^^. 

Proof:  Since  compactness  implies  perfection,  from  2.21  we  have 

The  other  part  of  the  theorem  follows  from  theorems  2.20  and  2.18. 

III.  The  Hensel-Klrschak  Field  36^. 

3.1.  The  Field  BEj. — Given  a  field  g  we  build  a  system:  || 

D  3.1  I,  ^  [all  <p-^^'-^'  :  3  :  3^5  z'  <  i^  ' )  '  <p(i)  =  z]. 

Here  3  =  [f}  is  the  class  of  all  integers  i. 

Addition  of  elements  of  the  system  is  the  usual  addition  of  functions: 

D  3.2  ^i  +  ^2  =  (^i(i)  4-  ^2(^)  I  i)         {(pi,  (pi). 

Multiplication  is  thus  defined:  ^ 

D  3.3  ipnp2  =  (        X)       <P\{i\)(pi{i2)  I  *)  (^1,  <P2). 

<l,'<2lii  +  i2=i 

*  S,  p.  199. 

t  S,  p.  220. 

JOi,  p.  281. 

§  Oi,  p.  284. 

11  In  this  definition  it  is  obvious  that  when  <p  is  not  the  zero  function  the  i^'s,  in  the 
definition  have  a  maximum.  We  denote  this  maximum  by  j'(^)  or  v^ov  more  simply  by  v 
when  it  is  obvious  to  which  element  it  belongs.  Also  in  this  case  (p{v)  7^  z.  In  case  ip  is 
the  zero  function  such  a  maximum  does  not  exist;  we  denote  this  symbolically  by  regarding 
V  as  =0 . 

\  It  is  to  be  noticed  that  the  summation  in  this  definition  is  finitely  non-zero,  and 
hence  no  questions  like  convergence,  etc.,  are  involved;  for  by  D  3.1  both  i\  and  12  have  finite 
lower  bounds  for  which  the  functional  values  of  <p\,  V2  respectively  are  non-zero,  and  since 
i\  •\-  ii  =  i  the  upper  bounds  too  will  be  finite  and  for  values  greater  than  these  the  products 
of  the  functional  values  will  be  zero. 


GoKHALE :   Concerning  Compact  Kiirschdk  Fields.  309 

We  shall  hereafter  designate  the  system  in  D  3.1  with  addition  and 
multiplication  as  defined  in  3.2  and  3.3  by  Xj.  We  also  introduce  a  notation 
for  a  special  set  of  elements  of  36s. 

D  3.4  di  =  (u  at  i  and  z  elsewhere)         (i). 

We  now  prove  the  theorem  that  this  system  is  a  field : 

Thm  3.1  g  • )  •  3ef. 

Proof:  The  properties  1-6  and  hence  6o  in  D  1.1  are  easily  seen  to  be 
satisfied  from  the  fact  that  ^^  is  a  field ;  the  zero  element  is  in  this  case  the 
zero  function  (p(i)  =  z  (i).  As  for  the  property  7  we  shall  prove  its  equiva- 
lent 7'.  The  first  part  of  this  property  is  obvious;  also  the  element  do 
satisfies  the  first  condition  for  the  unit  element,  and  in  fact  also  the  second 
condition,  that  is, 

(p  ^  2*5  •  )  •  ^(^'  ^  ipip'  =   5o. 

Consider  the  v  belonging  to  this  (p  (footnote  3.1) ;  then  an  effective  function 
(p'  is  the  function  (p'  with  (p'{i)  =  z  f or  ?  <  —  v,  (p'{—  v)  =  ul(p{v),  and 
<p\i)  f or  ?■  =  —  V  -{-  s  (s  >  0)  as  the  elements  of  i^  obtained  by  the  sequen- 
tial solutions  f or  5  =  1,  2,  •  •  •,  of  the  equations: 

X;  <p{v  +'ii)<p'i-  V  +  ^2)  =  2         {s>  0). 

Note  1:  If  we  denote  symbolically  hi  by  a:',  the  elements  of  the  field 
BEj  can  be  symbolically  expressed  as  infinite  series  J^ifiX\  where /i  =  (p(i)  (i). 
Addition  and  multiplication  can  then  be  regarded  as  the  corresponding 
formal  operations  on  the  infinite  series.  We  shall,  therefore,  hereafter  take 
these  infinite  series  as  the  elements  of  3tj  and  operate  with  them  since  this 
procedure  is  formally  more  convenient.  We  shall  sometimes  denote  the 
range  of  summation  of  the  index  i  as  from  i'  to  oo,  i;  being  the  integer 
defined  in  footnote  3.1. 

Note  2:  We  now  identify  (in  the  sense  of  isomorphism)  this  field  with 
the  field  ^(a;)  of  Steinitz,  where  x  is  transcendental*  with  respect  to  in- 
putting the  various  powers  of  x  in  one  field  into  correspondence  with  the 
same  powers  of  x  in  the  other,  and  the  elements  of  ^  with  the  same  elements 
of  %  in  the  other,  we  see  that  ^5  d  ^^  [a;],  where  ^^[a:]  is  the  integral  domain 
obtained  by  adjoining  the  transcendental  x.  Since  the  field  ^(a^)  is  obtained 
from  this  integral  domain  by  the  process  of  building  up  of  quotients,  f  we 
see  that  365  3  %(.x).  On  the  other  hand  it  is  obvious  that  3Cj  c  ^(x). 
Hence  the  result  required.  • 

*  S,  p.  184. 
t  S,  p.  178. 


310  Gokhale:   Concerning  Compact  Kilrschak  Fields. 

We  shall  therefore  now  speak  of  this  process  of  forming  365  from  f^  as 
the  adjunction  of  a  transcendental  element  x  to  the  field  %. 

3.2.  The  Hensel-Kiirschak  Field  ^Cj..— Let  ^  be  a  real  number  0  <  ^  <  1. 
We  now  define  the  modulus  for  the  field  H-^: 

2)3.5  Ih-'HNO;      <p9^zh')'\\^\\^^^'.. 

The  modular  properties  1  and  4  of  D  1.6  are  obvious.  The  property  2  is 
also  obvious  in  case  one  of  the  factors  is  the  zero  function.  In  the  other 
cases, 

V{ipnp2)    =    V((pi)  +   V{<p2)        {(Pi,  (P2). 

Hence 

!|  <P1<P2  II   =   r*"*^^  =    ^K^)+Kv.)  =   ^K^)^.(..)  =    II  ^,  II  II  ^2  ||. 

Thus  2  is  proved  completely. 

Again,  for  every  (pi  and  (p2,  v{<pi-\-(p2)=  the  first  i  for  which  (pi(i)-\-(P2{i) 
is  not  zero  (if  there  is  any  such,  otherwise  ^1  +  ^2  =  2  and  the  property 
3'  is  obvious) ;  thus 

v{(pi  +  (P2)  =  the  smaller  of  v((pi),  v{(p2). 

In  this  case  since  0  <  ^  <  1,  we  have: 

^1-^2  : )  :  II  ^1  +  <P2  II  =  k"''^'-^-'  ^  gr{^''V\  e"^')  .-.  ^  f(II  <Pi  II,  II  <P2  ||). 
Thus  365  is  a  Hensel-Kiirsqhak  field  (cf.  D  2.4),  denoting  by  36^  the  field  3Ej 
with  the  modulus  as  defined  in  D  3.5. 

Thm  3.2  S  • )  •  3£f. 

Hereafter  we  shall  generally  denote  the  elements  of  36,^  by  h{hi,  ^2,  etc.). 
The  subscript  ^^  in  36 ^  shall  be  dropped  unless  it  is  necessary  to  bring  in 
evidence  the  field  from  which  365  is  formed.  The  elements  of  <5  in  36g 
(by  isomorphism)  will,  when  it  is  desirable  to  bring  that  fact  in  prominence, 
be  denoted  by /(/i,/2,  etc.). 

3.3.  Perfection  of  365. — We  now  prove  the  theorem: 

Thm  3.3  S  • )  •  aef . 

Proof:  Consider  a  Cauchy  sequence  {A„}.  From  property  3'  of  the 
modulus  either  X„  1 1  ^n  1 1  =  0  in  which  case  the  limit  of  the  sequence  is  z  or 

3{no,v)^\\hn\\  =  r  =  -^ll^nll  in>  no) 

for  otherwise  if  e  <  X  1 1  ^^  1 1  which  exists,  and  a  =  jL  1 1  A„  1 1  —  e, 

n  >  Tie  :  )  :  3(wi,  ng)  ^  ||  hn,  \\  7^  \\  K^  \\')'\\  h^  -  h„,  \\ 

=  9ri\\hn,\\,\\hn,\\)>a 


Gokhale:   Concerning  Compact  Kiirschdk  Fields.  311 

where  n^  is  the  Ue  for  the  original  Cauchy  sequence,  and  this  is  impossible. 
With  this  V  and  A„  =  IIi""/n,  i  x'  (n),  we  have 

3no:^  :n>  no')  ' fn,  i  =  /no.  i  (1) 

for  otherwise  n  \)  :3  (ni,  n^)  >  n^\\  hn^  —  hn^  \  \  ^  ^""^  which  is  not  true 
as  {hn}  is  a  Cauchy  sequence.     Similarly, 

i  :)  :  3no,  i  :^  :  n  >  no,  i' )  '  fn,  ^i  =  fn,,  ^  v+i-  (2) 

If  we  now  define  h  =  J^i'"fiX\  where  fi  ^  /„„_  ._^,  i  {i  ^  v)  in  (1)  and  (2), 
we  see  that  i  :)  :  \\  h  —  h„^^  <  !l  =  ^"'^^  and  so  h  =  Lhn. 

n 

3.4.  Conditions  for  Compactness  of  36,ij. — Though  every  X  as  we  have 
seen  is  perfect,  it  is  not  necessarily  compact.  The  necessary  and  sufficient 
conditions  for  compactness  are  given  by  the  following  theorem: 

Thm  3.4  If  •  ~  •  %'">'  •  ~  •  5«"'t^ 

Proof:  Since/  ?^2')*||/||  =  1,  Lfn  =  h'  )  '  h^;  therefore  if  we  have 

a  Cauchy  sequence  of  elements  of  %  its  limit  element,  when  it  exists,  must 
be  an  element  of  %.  Take  now  an  infinite  sequence  of  elements  of  ^. 
1  is  an  upper  bound  of  the  modulus;  therefore  by  3^1"*  this  sequence  has  a 
Cauchy  subsequence  with  a  limit  element.  This  limit  element  by  the 
above  is  in  5.     Hence, 

It  is  also  obvious  that  g*^""^  •  )  •  ^<=pt^  f^j,  jj^  ^j^jg  (,^gg  every  infinite 
sequence  of  elements  of  ^  must  have  at  least  one  of  its  elements  repeated 
an  infinite  number  of  times. 

Next    suppose    g'"*;    consider    {hn}  ^  B  \\  hn\\  <  ^ ,    say    <  ^'.     Let 

n 

hn^  Y,fn,  i^:'  (n).  Then  jn,  i=  z  {n,  i  <  t).  Since  every  sequence  in  % 
has  1  as  an  upper  bound  of  the  modulus,  and  hence  by  S*'^*  has  a  Cauchy 
subsequence  with  a  limit  element  the  sequence  {/„,  <}  has  a  Cauchy  sub- 
sequence with  a  limit  element.  Take  the  corresponding  subsequence  of 
{hn}.  If  the  coefficients  in  this  sequence  be  denoted  by  letters  with  the 
upper  subscript  1,  we  have  a  sequence  whose  terms  begin  with  a  power 
not  less  than  t  and  the  coefficients  of  x*  form  a  Cauchy  sequence.  Also 
since  every  non-zero  element  of  ^  has  the  modulus  1,  after  a  finite  number 
of  terms,  the  terms  in  this  coefficient  sequence  are  identical.  The  coefficient 
sequence  {fn^t+i}  has  similarly  a  Cauchy  subsequence.  Take  the  corre- 
sponding subsequence  oi  {hn}.  If  we  continue  in  this  manner,  and  denote 
the  successive  distinct  subsequences  of  {hn}  obtained  in  this  manner  by 
lei,  lei,  '--Ah^'},  --^wehave 

{hn}  z>  {h'i^  D  {e-^^M  (r>0)  (1) 


312  Gokhale:   Concerning  Compact  Kilrschdk  Fields. 

and 

r>  0:)  :3nr:^:m>  r-n>  nr-i  <t-\-r\):ffi  =  jn^,  i-        (2) 

If  this  series  of  subsequences  has  a  last  term,  say  {hn^],  then  its  coefficient 
sequences  are  all  Cauchy  sequences  and  it  can  be  easily  proved  that  l^*"^} 
is  itself  a  Cauchy  sequence,  and  therefore  by  Xf,  with  a  limit  element  in  Xj. 

If  the  series  of  subsequences  has  no  last  term,  take  the  subsequence  of 
{hn}  formed  in  the  following  manner:  from  each  term  of  the  infinite  series 
select  the  first  element  which  does  not  belong  to  the  next  term  in  the 
sequence  (such  elements  do  exist,  since  the  terms  of  the  infinite  series  are 
distinct).  By  using  (2),  this  sequence  can  be  seen  to  be  a  Cauchy  sequence 
and  therefore  by  the  perfection  of  36 j,  with  a  limit  element  in  36 j. 

Finally,  to  prove  ^""^^ ' )  '  g^^te.  y^g  prove  this  by  proving  directly 
the  contrapositive,  viz., 

Cfclnflnite  •  \  .  ^— cpt  ^ 

For  consider    {fnV''''^'';    then  5  ||/n  ||  =  1.     If    this     sequence     has     a 

■Cauchy  subsequence,  e-)'3  (ni,  n2)  ^  \\fn,  -  fn,\\  <  e;  but  by  {/„ }  ^''''^'' 
and  the  fact  that/  4=  z  *  )  *  1 1  /  1 1  =  1,  we  have  ni  4=  ^2  ' )  '  1 1  /jh  —  /nj  1 1  =  1, 
and  these  contradict  each  other  when  e  <  1.     Hence  the  result. 

Thus  the  theorem  is  completely  proved. 

Corollary:  Since  every  finite  field  is  of  the  type*  [p^[],  where  p  is  prime 
and  q  a  positive  integer,  we  have : 

3.5.  Subfield  R(^,  x)  of  365. — Since  rational  integral  functions  of  a 
transcendental  x  with  coefficients  in  %  form  a  subclass  of  36j,  viz.,  the  class 
of  all  finitely  non-zero  functions  on  ^  to  5,  with  z  as  the  functional  value 
for  negative  values  of  the  argument,  the  field  of  all  rational  functions  of  such 
an  ir  is  a  subfield  of  365.     Thus, 

Thm  3.6  %')•  %,  D  R(5,  x). 

Further,! 

ThmS.7  ^«°"«  °'  °'  '""^  *yp«  ^^  • )  •  3^5  3  R(^,  a:) . 

Proof:  The  proof  is  partly  suggested  by  Hensel's  proof  of  the  theorem 
that  the  field  of  the  p-adic  numbers  includes  properly  the  field  of  ordinary 
rationals.J  In  analogy  with  Hensel  we  proceed  to  show  that  under  the 
conditions  in  the  hypotheses  every  element  of  R(i5,  x)  corresponds  to  a 

*  M,  p.  220. 

t  Here  the  symbol  U  reads  "properly  includes." 

t  H,  p.  38. 


Gokhale:   Concerning  Compact  Kiirschdk  Fields.  313 

periodic  element  of  3£g.     A  periodic  element  is  defined  thus: 

D3.6       {h=ZM'^'"'^'":=:h:^:3(s,t>0)^i>  s-)'fi=   U^t, 

the  smallest  such  t  is  called  the  period. 

All  polynomials  are  obviously  periodic,  the  period  being  1  and  the 
repeated  element  2.  The  product  of  two  periodic  elements  is  easily  seen 
to  be  periodic.  Thus  it  is  sufficient  to  prove  that  the  reciprocal  of  a  poly- 
nomial ^  irreducible  in  i^  is  a  periodic  element. 

In  case  %  is  of  the  type  "iP^,  that  is,  the  absolute  algebraic  field  *  charac- 
teristic p,  ip  is  of  the  first  degree  in  x.  In  case  (p  =  fx,  f  being  a  non-zero 
element  of  ^,  the  reciprocal  of  (p  is  fx~'^,  where  /  is  the  reciprocal  of  /, 
and  is,  therefore,  obviously  periodic.  In  case  (p  =  fx  -\-  fi  where  none  of 
/, /i  are  zero,  its  reciprocal  Is  /J/(l  -\-f2x),  where  /J  is  the  reciprocal  of/i 
and/  =  /]/2.     But  1/(1  +  f2x)  equals: 

l-f2X+f2X'-    •••   +    (-    iyf2X'-^    ••• 

and  since /2  is  algebraic  with  respect  to  the  field  [0,  1,  2,  •  •  - ,  p  —  l"]  there 
exists t  a  positive  integer  n  such  that/^  =  1.  Hence  the  series  is  periodic. 
In  case  %  is  finite,  ^  is  a  Galois  Field  J  [_p^'].  If  the  polynomial  <p 
irreducible  in  ^  is  of  degree  n  in  x,  the  class  of  all  polynomials  V'  in  a:  with 
coefficients  in  %  falls  modulo  (p  into  p^^  classes  of  congruent  polynomials, 
and  hence  there  exists  §  a  positive  integer  m  such  that  a;*"  =  1  (mod.  <p), 
that  is  to  say,  there  exists  a  polynomial  xp  such  that  <p(x)4/{x)  =  x^  —  1. 
Thus: 

lf<p(x)  =  xPix)llipix)4^(x)']  =  ^{x)l{x-^  -  1)  =  -  ■i/'(x)  { 1  +  a;-  +  0:2-  +  . . . } 

and  is  therefore  obviously  periodic. 

3.6.  Adjunction  of  Elements  to  X,^  algebraic  to  }^. — We  shall  now  prove 
a  theorem  required  for  the  sequel : 

where  *S  denotes  a  system  of  elements. 

Proof:  We  put  the  elements  of  the  two  fields  into  isomorphism  in  this 
manner:  put  the  systems  %  and  S  in  one  into  correspondence  with  those  in 
the  other;  since  aS  is  algebraic  with  respect  to  '^,  this  correspondence  will 
persist  under  all  the  field  operations  like  additions,  etc.  Also  since  the 
moduli  of  these  elements  in  %  and  S  are  the  same  in  both  cases  the  iso- 

*  Cf.  1.2;  also  S,  p.  199. 
t  S.,  p.  251. 
J  M,  p.  220. 

§  This  can  be  proved  by  the  usual  methods  of  Galois  Field  theory  (cf.  L.  E.  Dickson, 
"History  of  the  Theory  of  Numbers,"  Washington,  D.  C,  1919). 


314 


GoKBLA-LE :   Concerning  Compact  Kiirschdk  Fields. 


morphism  so  far  is  complete.  Also  both  36,;<s)  and  36,^(>S)  are  extensions  of 
^(/S).  If  we  now  identify  the  element  x  and  its  moduli  iti  the  two  fields, 
we  see  that  both  36j<s)  and  ^^(S)  are  extensions  of  36,^  Now  36j(»S)  is  the 
smallest  extension*  of  jEj  containing  the  system  S.  Therefore  36,^s)  o  BEjCS). 
Further  every  element  of  ^^^g)  is  of  the  form  J^iSiX*,  where  the  elements  s 
belong  to  \^{S),  and  is  thus  an  element  of  %yiS).  Thus  the  theorem  is 
completely  proved. 

IV.  The  Properties  p,  A,  P,  opt. 

4.1.  A  Complete  Existential  Theory. — In  the  preceding  sections  we  have 
defined,  with  reference  to  Kiirschak  fields,  the  four  properties  p,  A,  P, 
and  cpt.  If  the  existence  or  non-existence  of  these  properties  in  this  order 
is  denoted  by  positive  or  negative  signs  in  that  order,  a  given  Kiirschak 
field  will  have  one  of  the  2^=  16  characters  (H — | — [-  +  ),  (+H — I — ). 
(H — I h),  ••',  i )•  In  this  section  we  develop  a  complete  exis- 
tential theory^  of  these  properties,  that  is,  we  prove  2^  propositions  stating 

for  each  of  the  2*  combinations  (+  +  +  +  ),    ••••>   ( )  of  these 

properties  whether  there  exists  or  does  not  exist  a  Kiirschak  field  having 
the  particular  combination  of  the  properties.  These  propositions  are 
tabulated  below: 

Thm  4.1-4.16: 


Cases. 

P- 

A. 

.P. 

cpt. 

Consistent. 

2'.'.'.'.'.'.'.'.'.'.'. 

3 

4. 

5   . 

1 1 1 1 1 1 1 1 ++++++++ 

1  1  1  1  ++++  1  1  1  1  ++  +  + 

++ 1 1 ++ 1 1 ++ 1 1 ++ 1 1 

1  +  I  +  I  +  I  +  I  +  I  +  I  +  I  + 

f  +  1  +  1 

6 

+ 

7 

8... 

+ 

9   

+ 

10 

+ 

11 

12 

+ 

13 

+ 

14 

+    - 

15 

16 

+ 

Note:  The  +  and  —  entries  in  the  last  column  denote  the  existence  and' 
non-existence  respectively  of  a  Kiirschak  field  with  the  character  denoted 
by  the  entries  under  the  first  four  columns. 

Also  in  looking  the  entries  under  p,  it  is  necessary  to  remember  that  the 
entries  in  the  final  column  do  not  depend  upon  any  special  choice  of  the 


*  S,  p.  186. 
t  E.  H.  Moore. 
Press,  p.  82. 


'  Introduction  to  a  Form  of  General  Analysis,"  1910,  Yale  University 


Gokhale:   Concerning  Compact  Kiirschdk  Fields.  315 

prime  p.     Thus  the  entries  with  +  sign  under  p  we  interpret  as  p  * )  '  *3 
^^•^^',  and  those  with  -  sign  as  =^3  ^i"-"*"'. 

4.2.  Proofs: 

(a)  3,  7,  11,  and  15  follow  from  theorem  2.1,  viz.,  ^''p*  * )  *  9?^. 
(6)  1  follows  from  theorem  2.4,  viz.,  9=^^'°^* ' )  '  ^°. 

(c)  The  following  number  systems  in  analysis  with  the  absolute  value 
as  modulus  prove  9,  12,  13,  and  16  respectively: 

(9)  complex  numbers, 

(12)  algebraic  numbers, 

(13)  real  numbers, 
(16)  rational  numbers. 

(d)  To  prove  2  :  consider  (X,^^)(Apy,  this  has  the  properties  p,  A,  P. 
By  theorem  3.4,  36^p  is  P  and  —  cpi.  Using  2.15  we  see  that  (X>^)(4p) 
is  —  cpt 

(e)  To  prove  5:  consider  36 [pj.  This  field  is  ~A,  for  the  equation 
y^  —  X  =  0  has  no  solution.     However  by  3.4  it  is  cpt. 

( / )  We  get  6,  an  example  being  2i^p.    The  proof  is  similar  to  the  above. 
(g)  Similarly  36r  where  R  =  |^all  ordinary  rationals]  gives  us  14. 

4.3.  Proof  of  10:   We  proceed  to  prove  that: 

Thm  4.IO0  ©up^ 

where  <S  is  the  field  of  Hensel's  p-adic  numbers. 

Take  an  element  of  ®  whose  modulus  a  is  >  1.  Take  an  infinite 
sequence  of  positive  distinct  ordinary  rationals  {tn]  so  that  they  all  lie 
between  two  positive  rationals  r,  r' .  Let  rn  =  Inffnn,  where  In,  fUn  are 
positive  and  relatively  prime.  Consider  the  sequence  {hn}  where 
hy  -  Afn  =  0  (n).  Then  ||  hn  \\  =  a^»  (n)  .'.  {hn}^'''^'\  This  sequence 
an  upper  bound  of  whose  modulus  is  gr(a^,  a^'),  cannot  have  any  Cauchy 
subsequence. 

For  otherwise, 

e  :)  :3  ne  ^{nu  ni)  >  ne ' )  '  \\  K,  —  K^W  <  e, 
but  we  have 

ni  +  n2  : )  :  II  K,  \\  +  ||  K,  ||  :  )  :  ||  hn,  -  hn,  \\  =  gr{\\  hn,  \\,  \\  hn,  ||) 


> 


and  taking  e  <  a"",  we  get  a  contradiction.  . 

Hence  the  theorem. 

4.4.  Proof  of  4:  Consider  OC[p])a;  36[p]  is  perfect  by  3.3.  If  we  now 
prove  that  algebraic  closure  is  in  this  case  obtained  by  the  adjunction  of 
an  infinite  number  of  elements,  then,  by  2.21  (3iip{)'2^  and  hence  by  2.1 


316  Gokbale:   Concerning  Compact  Kiirschdk  Fields. 

—  cpt.  Now  (X[p])a  contains  3t.^  which  by  theorem  3.8  can  be  regarded 
as  obtained  from  X[p]  by  adjoining  algebraic  elements  to  [p].  The  number 
of  these  is,  however,  infinite ;  otherwise  ^^  would  have  only  a  finite  number 
of  elements.  Thus  {2cip])a  is  infinite  with  respect  to  96[pj  and  so  the  theorem 
is  proved. 

4.5.  Proof  of  8:    Ostrowski  in  his  theses*  has  proved  the  following 
theorem : 
Thm  4.8o  di'' '  »S«'«'  ^=  '"""^^  • )  '  ^(S) -^. 

We  use  this  theorem  to  prove  4.8. 

Consider  36[p],  p  >  2;  let  r  be  any  one  of  the  integers:  2,  3,  •  •  •,  p  —  1. 
From  Steinitz's  paper,  f  we  have 

^prlme.  >  p  •  \   .  (yq  A^  ~\  Irreducible  In  [p].  Ist  kind 

Consider  a  sequence  of  primes,  {g„}  such  that 

(1)  qi>P 

(2)  Qn  >  p^i^^^s  -   «"-! 

and  the  corresponding  infinite  system  of  irreducible  polynomials 

S^  (y^n-^r\n). 

Consider  now  j6[p](<S).  By  theorem  3.8,  3lt[p](/S)  =  3£[p](s).  Also  S  is  of 
the  first  kind,  further  if  Sm  ^  (g«»  ■^r\n=  1,  2,  •  •  •,  m  -  1),  MC^J 
contains  elements  whose  order  is  at  most  p^i'z  •••  9m-i  and  hence  the  poly- 
nomial y^vi-\-  r  is  irreducible  in  \jp^{Sm)  since  the  order  of  its  roots  is 
qM>  the  highest  order  in  \jpJi{Sn,)-  Thus  5  is  a  progressive  J  system.  We 
have  further  'X,[p]{S)~^,  since  the  polynomial  y"^  -{-  x  for  instance  has' still 
no  root.    Thus  X,,]  (5)^-^-^-"^*. 

*  Oi,  p.  280. 
1 8,  p.  231. 

tS,  p.  271. 


VITA. 

I,  Vishnu  Dattatreya  Gokhale,  was  born  in  Poona  City,  India,  on  the 
twenty-fourth  of  February,  1892.  I  got  my  secondary  and  high  school 
education  at  the  Nutan  Marathi  Vidyalaya,  Poona  City.  I  entered 
Fergusson  College,  Poona  City,  in  1907,  and  did  all  my  undergraduate 
and  part  of  my  graduate  work  there.  My  instructors  in  mathematics 
were  Prin.  R.  P.  Paranjpye  and  Professor  V.  B.  Naik.  I  got  the  degree  of 
Bachelor  of  Arts  in  1911,  and  Master  of  Arts  1913,  at  the  University  of 
Bombay.  During  1914-1917  I  worked  as  Professor  of  Mathematics  at  the 
Fergusson  College,  Poona  City. 

I  came  to  the  United  States  in  December,  1917.  During  1918-19  I 
took  courses  at  the  University  of  California  under  Professor  Cajori,  and 
Drs.  McDonald,  Buck,  Sperry  and  Bernstein.  I  joined  the  University  of 
Chicago  in  autumn,  1919,  and  during  nine  quarters  took  courses  in  mathe- 
matics under  Professors  Moore,  Bliss,  Dickson  and  Wilczynski,  the  major 
part  of  my  work  being  under  Professor  Moore. 

This  paper  was  written  under  the  direction  of  Professor  Moore  and  I 
wish  to  express  my  appreciation  of  his  constant  encouragement  and  in- 
valuable criticism  while  engaged  in  the  investigation. 

I  wish  to  acknowledge  my  indebtedness  to  all  the  men  under  whom  I 
have  done  my  graduate  work,  I  must  mention  Professors  Paranjpye  and 
Naik,  who  created  in  me  an  interest  in  mathematics,  and  Dr.  Sperry,  who 
advised  and  inspired  me  to  go  to  Chicago.  I  cannot  adequately  express 
what  I  owe  to  Professor  Moore.  While  attending  his  lectures,  and  still 
more  in  personal  interviews,  I  enjoyed  some  of  the  best  moments  in  my  life 
and  it  is  he,  more  than  any  one  else,  who  initiated  me  into  the  poetry  of 
mathematics  and  made  me  realize  so  vividly  that  mathematics  is  an  art. 


52n;?9- 


UNIVERSITY  OF  CAIIFORNU  UBRARY 


